Download What Cell Size Does the Computed Slope/Aspect Angle Represent?

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WhatGellSizeDoesthe
AngleReptesent?
Slope
Gomputed
/ Aspect
Michael E. Hodgson
Abstract
The computation of slope and aspect angl.esfot a ceL|is a
common procedure in environmental studies and remote
sensing applications in which topography is impofiant.
WhileThe algorithm for computing slope/aspect angles rcquires eithei four or'eight neighbors in a centered thtee by
ihree window of cells.lhe estimated angles are used as if
they depict the'surface orientation of only the single central
celi. Two questions result from this observation. What cell
size does ihe slope and aspect angle derived from this win- dow best represent?How different is the actual surface angle
of the cential cell from the surface angle comP-utedusing the
window of elevation valuesTAlthough this difference in computation i"rsus useis somewhatknown, it has never been
-documented.
This article empirically demonstrates that the
slope/aspect angle derived from the neighboring elevation
points best depicts the surface orientation for a \arye-rcell'either
1.6 times or 2.0 times larger than the size of the central ceLl.It is suggestedthat, rather than first tesampling elevation datasets of a finer resolution to a larger cell size
commensurate with other data in a study and then deriving
ilope/aspect angles, a mean slope/aspect angular measurc'
ment be'derived directly from the higher resolution data for
each larger cell size.
lntroduction
Surfaceslope and aspectare commonly used by-produc-tsof
an elevation surfacefor a wide variety of applications. Sometimes the elevation grid used is a resampling from a finer elevation grid to a coarserelevation grid so that the cell sizes
of differ6nt data layers are commensurate' This resampling
process may be thiowing away information important in the
ilope/aspeit computation. The computation of slope/aspect
for-each-surfacec-ellis made from some number of neighboring elevation values in a three by thr,eewindow but is used
as"if it representsthe surfaceanglesfor only the central cell'
It is ofte; assumedthat the computed surfaceanglesactually
representa cell size twice as large as the original grid cell.
Although this difference in computation versus use is somewhat kiown, the representative ceII size has never been documented. Further, the magnitudes of error which exist
between the use of the angle for a one cell area and the actual angle for that cell are not well understood.In other
words, are the angles all that different?
Using a well-known synthetic surface, this pap-erempirically detJrmines the cell size that the bi-directional surface
normal computed from elevation values in a three by three
window actually represents.Three different slope/aspectalgorithms are examined with three different grid-cell sizes.
Oak Ridge National Laboratory,Building 4500N' MS 6274,
Oak Ridge,TN 37831-6274.
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The results indicate that the estimatedsurfacenormal from
the four nearestneighboring cells in a three by three window
best models the surlace of i cell 1.6 times as Iarge as the
samplins interval in the elevation grid' For slope/aspectalgorithirs u"singthe eight nearest neighbors,in a three by three
window, thE actual-cell representedis about 2'0 times the
sample interval. These relitionships are consistentregardless
of the sample interval in the original elevation grid'
Methodologly
The size olihe grid cell portrayed by the elevation surface
samplins intervil and the cell iize that is best representedby
the istiriated angle from a 3 by 3 matrix of elevation values
will be hereafterleferred to as the grid-cell size and representative cell size, respectively. Determining the repre-sentative cell size will invo^lvea continuous range of candidate
representativecell sizes.The examination between grid-cell
siie and representativecell size was conducted using amathematiially defined synthetic terrain su-rface'A mathematically defiried surface-allowsperfect definition of the elevation poi.ttt ott the surface and near-p,erfectlestimation of
the mean surfacenormal of any size cell on the surface'To
demonstratethe difference in ingles estimatedby slope/asoect alsorithms using different numbers of neighbors,three
used algoiithms were tested.Becauseestimation
"o*-o."nly
of surfaceproperties is sensitiveto the sampling-intensity,
three diffeienf grid-cell sizes were also examined'
TheTestSudace
This study used the 4g-term trigonometric surfaceby Monison (197i 1974) as the synthetic terrain surface to compare
each-representationcel iize and algorithm-(Figurer)' The
srrrface'isa complex repeating seriesof-undulations with a
maximum slopebf 60.48 degrees.Basedon a preliminar{ 9xamination of ihe surface,cell sizes of sO by 50, 100 by 100'
and 200 by 200 units were selectedto representdifferent
sampling intensities.
ftr"it,t" surfacenormal for the grid cell was computed
using a pseudo-integrativeapproach-lhe vector mean surface"normal of a fin6r sampl-e-ofthe elevation surface within
each cell. This subsample1s a 21,by 21'matrix of elevation
lThe estimation of the true surfaceslope/aspectis "near-perfect" becauseit is approximated with a pseudo-integrative
logic.
PhotogrammetricEngineering& Remote Sensing,
V o l . 6 r , N o . 5 , M a Y 1 9 9 5 'P P . 5 1 3 - 5 1 7 .
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1995
O
and Remote Sensing
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The aspect of the surfacewas computed from an algorithm
using the change in elevation values in the we and sn gradients (seealgorithm in Ritter (1987)).
The bi-directional surfacenormal of a cell may be represented as a vector of unit length in x-y-z hemispherical
space:i.e.,
fx,y,zl
Figure1. A perspectiveview of the syntheticsurface
used to compareactual and estimatedsurfaceorientation. The surfaceis generatedby using a 49-termtrigonometric series after Morrison(1971,;1.974).The original
surfacefrom Morrison'swork was rotated 50 degreesto
avoidbias in the ridgesrunningeast-west.Tick marks
are at a 2OO-unit
spacing.
values (Figure 2a). Each subcell in the 21 by 21 subsampleis
1,121,th
the size of the candidate representativecell size under investigation. (The 1121thsubsampleis based on an empirical test that found a negligible difference in angular
measurementsusing a smaller subcell size.) As an example,
441 subcellsof Z.gAby 2.38 units in size were used to estimate the mean surfacenormal of a cell 50 by 50 units in
size. When examining the 200- by 200-unit cell size, 441
subcellsof 9.52 by 9.52 units in size were used to compute
the vector mean.
To determine the cellsize representedbv the bi-directional surfacenormal derived from the 3 by 3 grid-cell submatrix, a comparison was undertaken between this angle and
the anglesfrom a continuous range of candidate representative cell sizes from 0.1 to 3.0 times the size (along one axis)
of the grid cell. Each candidaterepresentativecell was centered at the same location as the grid cell, and the pseudointegrative approach for computing the true surfaceangles
was performed. A random sample of so grid cells on the elevation surface (Figure 1) was used in determining the mean
angular error for each candidate representativecell size. The
representativecell size was taken as that candidate cell size
exhibiting the least mean angular error.
Bi-Directional
Surface
Nomd
To avoid the problems associatedwith undefined aspectof
"flat"slopes, bi-directional surfaceangleswere used in this
analysis rather than independent anglesof slope and aspect.
Bi- directional anglesare also better indicators of surfaceorientation for applications such as the estimation of solar
insolation or the topographic normalization of remotely
sensedimages.Slope/aspectis computed from the normal
vector of a plane surfacebased on the cross products of two
vectors in orthogonal axes on the surface.The slope in two
orthogonal gradients,typically west-to-east(we) and south-tonorth (sn), are used for this estimation: i.e.,
Slope' : Tan' (VSlope|,+Slope?")
(1)
where x : sin aspecf * sin slope
y : cos aspect * sin slope
z -- cos slooe
For those instancei when the slope of an observationis 0.0
(and, thus, the aspectangle is "uhdefined"), the coordinate
locations for the vector endpoint are set to x : 0.0, y: 6.9,
and z : 1.0. This allows "flat" surfacesto be included in the
computation of the vector mean.
VectorMeanSufaceNomal
An algebraicaverageof bi-directional surfacenormal measurements from a number of surface angle observations is not
possiblebecauseaspectanglesare measurementson a circular scale,not a linear scale.However, after work in other
fields on directional statistics (Rayleigh,1BB0;Watson, 1966;
Agterberg,1974; Gaile and Burt, f OB0),the vector mean was
used as a "mean" surfaceorientation. The vector mean reDresenting the surface normal in this 21 by 21,matrix of subcells was computed by adding the vectoi endpoints for each
observation:i.e.,
Nreprasantative
:
(3)
Enorin Surface
NomalAngle
The bi-directional angular difference (bi.), or angular error,
between the surfacenormals for a candidate celf size and the
grid-cell size was derived from
bi.: cos'f- -i*-*"''''*'il=' I
*
1N,"o."."n,u,,,"
|
lN..rr l-
Slope/Aspect
Algorithms
The fundamental differencesbetween most slope/aspectalgorithms is in the number of neighboring cell values used and
the weighting of each cell value. The most common algorithms use either four or eight of the neighborsin a thiee by
three window centeredon the cell in question (Figure 2b).
When using all eight neighbors, variatibns in algoiithms use
different weights for the diagonal neighbors.This study used
one algorithm employing four nearestneighbors,a finite-differencealgorithm using eight nearestneighbors,and a regression plane fitted to the eight nearestneighbors.
A commonly used algorithm that estimates surface angles from only the four nearestneighboring elevation values
in the grid was suggestedby Fleming and Hoffer (rgz9) and
presentedin algorithmic form by Ritter (1987):i.e.,
Slope", :
: ++],
slope*,
(5)
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ARIICLE
I- srourpr"
f
|I
Ceilsize
(121thof
@Isize
undet
inwstigatiil)
b)
a)
betweenthe "cell" impliedby the samplingresFigure2. Relationship
olution of the grid (i.e., the grid celt)and the size of a candidalerepresentativecell in (a). The relativesize of lhe representativecell in this
1.6 times the grid cell. Notationfor the
illustrationis approximately
elevationvaluesin a centeredthree by threewindowis
neighboring
s h o w ni n ( b ) .
where e, : elevation value of the 3 by 3 submatrix (Figure
2b).
A third-order finite difference method using eight neighboring elevation values (by differencing the set of elevations
on opposite sides of the central cell) was suggestedby
Sharpnackand Akin (1969):i.e.,
(e'*eo-|eu)- (gu*e'*e')
:'
sloPe""6*cell size
: ,*."';:"j;;lfj*.*,
srope"*
(6)
The method suggestedby Sharpnack and Akin produces the
same results as a multiple linear regressionmodel (or a leastsquires fitted plane to the eight elevation values),yet is computationally more efficient.
Horn (1sar) presenteda modified version of Sharpnack
and Akin's method using unequal weights for the closer elevation values:i.e.,
_
or^-^ =
)IoPe-n
b t o P e - , ':
( e , - t 2 e o *e ) - ( e o * 2 e r + e , ) .
'
B.*Il drr"
(eu* 2 e, * er) (e, + 2e, * eo)
S.*ll
(7)
,ir"
andlmplications
Results
The results of the tests indicate a direct and stable relationship between the grid-cell size and the representativecell
size. The representativecell size was found to be either 1.6
or 2.0 timeJthe grid-cell size, depending on the number of
neighborsused in the slope/aspectalgorithm (Figure !). the
surfacenormal produced from the algorithm that used only
four neighboring values was most closely associatedwith a
representativecell about 1.6 times the grid-cell size (Figure
4): The other two algorithms' using eight neighborswere
cell size-ap-_
most closelyassociatedwith a representative
proximately 2.0 times the grid-cell size. This relationship be'The regression algorithm produced angles that repres-ented a
cell size slightly larger-close to 2.1 times the grid-cell size.
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tween algorithm and representativecell sizeswas consistent
regardlesi of the grid-cell size (or relative surfacesampling inlensity). It would be expectedthat this relationship would
hold true for any study area and sampling intensity'
What may be surprising is (1) the magnitude of error between the repiesentativecell size and grid-cell size surface
anelesand (j) tfre relative accuracybetween algorithms. The
,uJ.us. bi-directionalangularerror betweenestimatedand
true s]urface
anelesat tht1.0 cell size (i.e.,the cell size such
measuresare uied to represent)ranged from 0'74 to 1.51,
2.30 to 4.68,and 7.45Io 9.50 degreesfor the 50-, 100-,and
200-unitcell sizes,respectively(Figure3 and Table r). The
low mean angular error for the smaller cell sizes was expected and iridicates the similarity of surfaceanglesin close
proximity to each observation.If ihe surfaceangles derived
irom the-3 bv 3 window were actually used for the representative cell siie (1.6 or 2.0 times larger cell sizes)the average
error would only rangefrom 0.48 to 0.65, 1"37to 1.93,and
1.91 to 4.08 degiees.Thus, it is suggestedthat the error in
surfacenormal anglesmay be reduced by a factor of from
two to four by computing surfaceangleswith finer resolution
data but using these computed anglesto representcoarser
resolution daia. Obviously, the magnitude of error could be
lesseror greaterfor other data sets and cell sizes.
The algorithm using four neighboringvalues was consistently morJ accurate foi estimating-surface angles at the Sridcell iize (i.e..the 1.0 candidatecell size).For instance,the
grid-cell size was
mean angular error at the 50- by SO-unitsonly 0.7{ degreesfor the four neighbor algorithm but was
1.26 and 1.5i degreesfor the two eight neighboring algo- -rithms. This finding was true for each of the three grid-cell
sizes examined. Thus, in the typical application where the
surfacenormal anglesare used to representthe grid-cell si3e,
the algorithm using only four neighborsshould be used' This
findin"gis in contrist to the wg$by Skidmore (1989),who
deterriined that the eight-neighbor algorithms were more accurate than the four-neighboralgorithm. The methodology
used by Skidmore for defining truth, however, included an
estimaiion of "true" slope/aspectanglesfrom a contour-map
where all eight neighboiing values were used to manually estimate surfaceangles.
One implication of this study would suggestthat eleva-
PEER.REVIEWED
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al
10
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f
tr
l|l
E,
J
f
o
z
z
ut
o
z
z
=
UJ
=
a)
c)
d)
Figure3. Relationshipbetweenbi-directionalangle errorfor 5O-,1OO-,and
2OO-unitgrid-cellsizes and candidaterepresentativecells centeredat the
same locationand increasingin size.Thethree slope/aspectalgorithmsexaminedare in (a) through(c).A graphicaloverlayof the mean angularerror
for the 200-unitgrid-cellsizes for each algorithmis shown in (d).
tion data be collected at a higher spatial sampling intensity
than the other data layers in an application. In other applitations, where the elevation data must be eeneralizedto t
coarsersample size, it may be desirableiand likely more accurate) to determine the slope/aspectanglesfor the larger
cells from a vector mean of the smaller cell sizes. Slope/aspect anglescould be derived for the grid-cell size of this
greatersampling intensity, and then the slope/aspectsurface(s)could be resampledto the coarserreiolution of the
other data layers using some vector mean interpolation
method. For instance,rather than using 30- bv 30-m elevation data to derive slope/aspectangles"fora siudy that also
uses-30-by 3O-mThematic Mapper imagery, one might use
an elevation layer sampled at S m by 5 m or 15 m by fS m.
Slope/aspectanglesrepresenting30- by 30-m cell siies may
be computed utilizing a vector mean from an elevation grid
of S- by 5-m cell sizesduring the resampling process.However, other empirical tests are required to determine appropriate methods for weighting each observationand for
developing an interpolation function. The selection of the
ideal cell size, however, is problematic becauseone often relies on available elevation data, such as U.S. GeologicalSurvey or DefenseMapping Agency derived data. The availabi516
lity of softcop_yphotogrammetryand stereophotography may
allow the analyst greater freedom to determine the sampling
Figure4. Relationship
between the grid€ell size of
the systematicsample of
the elevationsurfaceand
the grid-cellsize represented by the slope/aspect anglesderivedfrom
algorithmsusingeither
four or eight neighbors.
'*-^2.9luf
i
und @IBEe
+i
PEER.REYIEWED
ARTICTE
cations of Remote Sensing, Purdue University, West Lafayette,
Indiana.
Gaile, G.L., and J.E.Burt, 1980.Directional Statistics,University of
Flemming/
East Anglia, Norwich.
Regression
Horn's
Hoffer's
Horn, B.K.P.,1980.Hill Shading and the ReflectanceMap, ProceedCandidate
50 100 200
50 100 200
ings of the IEEE,69(1):7a-47.
50 100 200
Cellsize'
Morrison,
J.L.,1s71. Method-ProducedEnor in Isorithmic MqPPinS,
1
4
.
8
6
1
.
8
8
6
.
2
2
r.o7 3.76 72.42 1 . . 6 4 5 . 5 1 1 4 . 0 9
0.1
Monograph No. CA-S, American Congresson SurveyTechnical
1,.063.71 72.23 1.62 5.45 13.89 7.a7 6.17 74.66
o.2
Washington,D.C.
Mapping,
ing
and
1 . 0 4 3 . 6 2 1 1 . 9 1 1 . 6 0 5 . 3 7 1 3 . 5 6 1.85 6.08 14.34
0.3
StatisticalTrends in Various Interpolation
Observed
1974.
1.3.50
1,.82
5.97
5
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2
5
1
3
.
1
1
1
.
5
8
1
1
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4
9
1
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0
2
3
.
5
1
0.4
Algorithms Useful for First StageInterpolation, The Canadian
0.98 3.36 10.96 1.54 5.10 12.56 7.79 5.82 73.35
Cartograp h er, L7(2)2142-759.
0 . 9 4 3 . 1 9 1 0 . 3 6 T.50 4.92 11.52 7 . 7 4 5 . 6 4 7 2 . 7 1
0.6
0.90 2.99 9.69 1 . 4 5 4 . 7 1 7 1 . 2 0 1 . 7 0 5 . 4 3 1 1 . 9 9
o.7
L., 1880. On the Resultantof a LargeNumber of Vibrations
Ravleieh,
't.64 5.20 1.1..27
o . 8 5 2 . 7 8 8 . 9 7 t.39 4.47 70.41.
0.8
of"the Same Pitch or Arbitrary Pause,Pftilosophical Magazine'
0.80 2.54 8.22 7.33 4.22 9.57 1.58 4.95 10.37
0.9
7O:73-78.
o.74 2.30 7.45 1.26 3.94 8.69 1.51 4.68 9.50
1.0
Ritter, P., 1987.A Vector-BasedSlope and Aspect Generation
0.68 2.06 6.70 1 . 1 9 3 . 6 5 7 . 7 9 \.44 4.39 8.60
7.1.
Algorithm, Photogtammettic Engineering & nemote Sensing,
't.2
0.63 1.84 6.00 1 . 1 0 3 . 3 4 6 . 8 7 1.36 4.08 7.68
5 3 ( 8 ) :110 s - 1 1 1 .
0.58 1.66 5.37 1.02 3.03 5.96 7.27 3.77 6.78
1.3
D.A., and G. Akin, 1969.An Algorithm for Computing
Sharpnack,
5.O7 1.18 3.46 5.89
+.o3 o.g3 2.71
u,cJ
l.c+
L,4
Aspect fiom Elevations, Photogramnetic Engineering,
and
Slope
4.44
o.83 2.47 4.22 1.09 3.15 5.06
1.51
0.50
I.J
35(3):247-248.
4
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3
0
2
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8
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2
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1
7
3
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4
5
0
.
9
9
4.18
o
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7
3
7.54
o.48
1.6
Skidmore,A.K., 1989.A Comparisonof Techniquesfor Calculating
0.49 1.66 4.08 0 . 6 4 1 . 8 5 2 . 7 7 0 . 9 0 2 . 5 8 3 . 6 3
Gradient and Aspect from a Gridded Digital Elevation Model, Ino.54 1.84 4.77 0.56 1.63 2.23 o.82 2.33 3.09
1.8
ternational Jourial of Geographical Information Sysfems,3(4):
0.62 2.O9 4.25 0.50 1.45 1.95 o.75 2.13 2.72
1.S
2.57
323-334.
1.98
7.37
7.97
0.69
4.47
o.45
o.72
2.40
2.O
2 . O 5 0.65 7.93 2.58
o . B 4 2 . 7 4 4 . 7 7 o.45 t.44
2.7
Watson. G.S.. 1966' The Statistics of Orientation Data, /ournol of
0.98 3.13 5.0S 0.51 1.61 2.34 0.66 1.97 2.77
2.2
GeologY,74:787-797.
r.12 3.53 5.43 o.5g 1.87 2.69 o.70 2.70 2.93
2.3
7 June 1993; accepted14 September1993; revised 8 OctofReceived
7.27 3.95 5.77 o.71 2.2r 3.08 o.77 2.30 3.21.
ber 1993)
1 . 4 3 4 . 3 7 6 . 1 1 0.85 2.60 3.49 0 . 8 6 2 . 5 3 3 . 5 5
2.5
7.59 4.79 6.43 1.00 3.00 3.90 0.96 2.80 3.9L
2.6
1,.765.22 6.75 r.r7 3.41. 4.32 r.o9 3.72 4.30
2.7
Michael E. Hodgson
1.94 5.64 7.O7 7.34 3.A2 4.74 7.24 3.47 4.71
2.8
Michael E. Hodgson is a member of the research
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staff and is a Te-amLeader for the Geographic
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2.30 6.46 7.69 7.70 4.64 5.58 L.57 4.21
3.0
CELL
CANDIDATE
Ennon nr DIFFERENT
Teele 1. Melt HevtspHrRtcAlANGULAR
(ERRoRlN DEGREES)
SrzESFoRALcoRrnv ANDGRIICELLSlze CoMstNnrtoNS
'The candidate cell size is given as a ftaction of the regular tesselation in the elevation model. For instance,a candidatecell size of 1.6
at the 50- by 50-m grid-cell size is 8o by 80 m.
resolution (under the limitation of the quality of and base/
height ratio of the photography).
References
Agterberg,F.P., t974. Calculation of PreferredOrientationsfrom
"
Vect"orial Data, Geomathematics:Mathematical Background and
Geo-ScienceApplications, Elsevier Scientific Publishing Company, New York, pP. 475-508.
Fleming, M.D., and R.M. Hoffer, 7979. Machine Ptocessing of Landsa{MSS Data and DMA Topographic Data fot Forest Cover Type
Mapping, LARS Technical Report 062879, Laboratory for Appli-
Researchand Applications team in the GIS &
Computer Modeline Group at the Oak Ridge Nationai Laboratorieslomll). He holds a B.A. deuu
the M'S. and
and
r u both
g r e g I T o m ltie
ofr Tennessee
u
I ururvDDvE d
ulllver'srly
l l e University
Ffr.n. in Geographyfrom the University of South Carolina'
Prior to his posilion at oRNL,Dr' Hodgson was a faculty
member at the University of Colorado from 1987 to 1994'
where he researchedcomputational analysis problems and
human responsesto natural hazards' Also at Colorado,he
was the Principal Investigator on an NSr award that converted the GeographyDepartment'spc based-Glshemote
teaching/resear
a 1S-workstationteaching/research
lab to a-ts-workstation
sensinsteachirig
teachins lab-to
sensing
Iaboraiory.His ipecific researchinterestsinclude the use of
remote sensingand CtS methods for mapping and monitoring environmental changes'Current researchactivities inclide the characterizatidnof environmental hazards,regional
modeling of ecosystemprocesses'and knowledge-basedinformation extraction'
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